\(\int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx\) [1669]

Optimal result

Integrand size = 26, antiderivative size = 266 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}} \] Output:

2*(-c*f+d*e)^(1/4)*(-f*(d*x+c)/(-c*f+d*e))^(1/2)*EllipticPi(d^(1/4)*(f*x+e 
)^(1/4)/(-c*f+d*e)^(1/4),-b^(1/2)*(-c*f+d*e)^(1/2)/d^(1/2)/(-a*f+b*e)^(1/2 
),I)/b^(1/2)/d^(1/4)/(-a*f+b*e)^(1/2)/(d*x+c)^(1/2)-2*(-c*f+d*e)^(1/4)*(-f 
*(d*x+c)/(-c*f+d*e))^(1/2)*EllipticPi(d^(1/4)*(f*x+e)^(1/4)/(-c*f+d*e)^(1/ 
4),b^(1/2)*(-c*f+d*e)^(1/2)/d^(1/2)/(-a*f+b*e)^(1/2),I)/b^(1/2)/d^(1/4)/(- 
a*f+b*e)^(1/2)/(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=-\frac {4 \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt [4]{\frac {b (e+f x)}{f (a+b x)}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},\frac {1}{4},\frac {7}{4},\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{3 b \sqrt {c+d x} \sqrt [4]{e+f x}} \] Input:

Integrate[1/((a + b*x)*Sqrt[c + d*x]*(e + f*x)^(1/4)),x]
 

Output:

(-4*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*((b*(e + f*x))/(f*(a + b*x)))^(1/4)* 
AppellF1[3/4, 1/2, 1/4, 7/4, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/ 
(f*(a + b*x))])/(3*b*Sqrt[c + d*x]*(e + f*x)^(1/4))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {117, 116, 993, 1542}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx\)

\(\Big \downarrow \) 117

\(\displaystyle \frac {\sqrt {-\frac {f (c+d x)}{d e-c f}} \int \frac {1}{(a+b x) \sqrt [4]{e+f x} \sqrt {-\frac {d x f}{d e-c f}-\frac {c f}{d e-c f}}}dx}{\sqrt {c+d x}}\)

\(\Big \downarrow \) 116

\(\displaystyle -\frac {4 \sqrt {-\frac {f (c+d x)}{d e-c f}} \int \frac {\sqrt {e+f x}}{(b e-a f-b (e+f x)) \sqrt {1-\frac {d (e+f x)}{d e-c f}}}d\sqrt [4]{e+f x}}{\sqrt {c+d x}}\)

\(\Big \downarrow \) 993

\(\displaystyle -\frac {4 \sqrt {-\frac {f (c+d x)}{d e-c f}} \left (\frac {\int \frac {1}{\left (\sqrt {b e-a f}-\sqrt {b} \sqrt {e+f x}\right ) \sqrt {1-\frac {d (e+f x)}{d e-c f}}}d\sqrt [4]{e+f x}}{2 \sqrt {b}}-\frac {\int \frac {1}{\left (\sqrt {b e-a f}+\sqrt {b} \sqrt {e+f x}\right ) \sqrt {1-\frac {d (e+f x)}{d e-c f}}}d\sqrt [4]{e+f x}}{2 \sqrt {b}}\right )}{\sqrt {c+d x}}\)

\(\Big \downarrow \) 1542

\(\displaystyle -\frac {4 \sqrt {-\frac {f (c+d x)}{d e-c f}} \left (\frac {\sqrt [4]{d e-c f} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{2 \sqrt {b} \sqrt [4]{d} \sqrt {b e-a f}}-\frac {\sqrt [4]{d e-c f} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{2 \sqrt {b} \sqrt [4]{d} \sqrt {b e-a f}}\right )}{\sqrt {c+d x}}\)

Input:

Int[1/((a + b*x)*Sqrt[c + d*x]*(e + f*x)^(1/4)),x]
 

Output:

(-4*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*(-1/2*((d*e - c*f)^(1/4)*EllipticPi 
[-((Sqrt[b]*Sqrt[d*e - c*f])/(Sqrt[d]*Sqrt[b*e - a*f])), ArcSin[(d^(1/4)*( 
e + f*x)^(1/4))/(d*e - c*f)^(1/4)], -1])/(Sqrt[b]*d^(1/4)*Sqrt[b*e - a*f]) 
 + ((d*e - c*f)^(1/4)*EllipticPi[(Sqrt[b]*Sqrt[d*e - c*f])/(Sqrt[d]*Sqrt[b 
*e - a*f]), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e - c*f)^(1/4)], -1])/(2*S 
qrt[b]*d^(1/4)*Sqrt[b*e - a*f])))/Sqrt[c + d*x]
 

Defintions of rubi rules used

rule 116

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
1/4)), x_] :> Simp[-4   Subst[Int[x^2/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) 
 + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] 
 && GtQ[-f/(d*e - c*f), 0]
 

rule 117

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
1/4)), x_] :> Simp[Sqrt[(-f)*((c + d*x)/(d*e - c*f))]/Sqrt[c + d*x]   Int[1 
/((a + b*x)*Sqrt[(-c)*(f/(d*e - c*f)) - d*f*(x/(d*e - c*f))]*(e + f*x)^(1/4 
)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[-f/(d*e - c*f), 0]
 

rule 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

Maple [F]

\[\int \frac {1}{\left (b x +a \right ) \sqrt {x d +c}\, \left (f x +e \right )^{\frac {1}{4}}}d x\]

Input:

int(1/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/4),x)
 

Output:

int(1/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int \frac {1}{\left (a + b x\right ) \sqrt {c + d x} \sqrt [4]{e + f x}}\, dx \] Input:

integrate(1/(b*x+a)/(d*x+c)**(1/2)/(f*x+e)**(1/4),x)
 

Output:

Integral(1/((a + b*x)*sqrt(c + d*x)*(e + f*x)**(1/4)), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} {\left (f x + e\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/4),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)*sqrt(d*x + c)*(f*x + e)^(1/4)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} {\left (f x + e\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/4),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)*sqrt(d*x + c)*(f*x + e)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int \frac {1}{{\left (e+f\,x\right )}^{1/4}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/((e + f*x)^(1/4)*(a + b*x)*(c + d*x)^(1/2)),x)
 

Output:

int(1/((e + f*x)^(1/4)*(a + b*x)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\text {too large to display} \] Input:

int(1/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/4),x)
 

Output:

(4*(e + f*x)**(1/4)*sqrt(c + d*x)*f - 9*sqrt(e + f*x)*int(((e + f*x)**(3/4 
)*sqrt(c + d*x)*x**2)/(3*a**2*c*d*e**2*f + 6*a**2*c*d*e*f**2*x + 3*a**2*c* 
d*f**3*x**2 + 3*a**2*d**2*e**2*f*x + 6*a**2*d**2*e*f**2*x**2 + 3*a**2*d**2 
*f**3*x**3 + a*b*c**2*e**2*f + 2*a*b*c**2*e*f**2*x + a*b*c**2*f**3*x**2 + 
2*a*b*c*d*e**3 + 8*a*b*c*d*e**2*f*x + 10*a*b*c*d*e*f**2*x**2 + 4*a*b*c*d*f 
**3*x**3 + 2*a*b*d**2*e**3*x + 7*a*b*d**2*e**2*f*x**2 + 8*a*b*d**2*e*f**2* 
x**3 + 3*a*b*d**2*f**3*x**4 + b**2*c**2*e**2*f*x + 2*b**2*c**2*e*f**2*x**2 
 + b**2*c**2*f**3*x**3 + 2*b**2*c*d*e**3*x + 5*b**2*c*d*e**2*f*x**2 + 4*b* 
*2*c*d*e*f**2*x**3 + b**2*c*d*f**3*x**4 + 2*b**2*d**2*e**3*x**2 + 4*b**2*d 
**2*e**2*f*x**3 + 2*b**2*d**2*e*f**2*x**4),x)*a*b*d**2*f**3 - 3*sqrt(e + f 
*x)*int(((e + f*x)**(3/4)*sqrt(c + d*x)*x**2)/(3*a**2*c*d*e**2*f + 6*a**2* 
c*d*e*f**2*x + 3*a**2*c*d*f**3*x**2 + 3*a**2*d**2*e**2*f*x + 6*a**2*d**2*e 
*f**2*x**2 + 3*a**2*d**2*f**3*x**3 + a*b*c**2*e**2*f + 2*a*b*c**2*e*f**2*x 
 + a*b*c**2*f**3*x**2 + 2*a*b*c*d*e**3 + 8*a*b*c*d*e**2*f*x + 10*a*b*c*d*e 
*f**2*x**2 + 4*a*b*c*d*f**3*x**3 + 2*a*b*d**2*e**3*x + 7*a*b*d**2*e**2*f*x 
**2 + 8*a*b*d**2*e*f**2*x**3 + 3*a*b*d**2*f**3*x**4 + b**2*c**2*e**2*f*x + 
 2*b**2*c**2*e*f**2*x**2 + b**2*c**2*f**3*x**3 + 2*b**2*c*d*e**3*x + 5*b** 
2*c*d*e**2*f*x**2 + 4*b**2*c*d*e*f**2*x**3 + b**2*c*d*f**3*x**4 + 2*b**2*d 
**2*e**3*x**2 + 4*b**2*d**2*e**2*f*x**3 + 2*b**2*d**2*e*f**2*x**4),x)*b**2 
*c*d*f**3 - 6*sqrt(e + f*x)*int(((e + f*x)**(3/4)*sqrt(c + d*x)*x**2)/(...